A dot planimeter is a device used in

planimetrics Planimetrics is the study of plane measurements, including angles, distances, and areas. History To measure planimetrics a planimeter or dot planimeter is used. This rather advanced analog technology is being taken over by simple image measurement ...

for estimating the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...

of a

shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...

, consisting of a transparent sheet containing a square grid of dots. To estimate the area of a shape, the sheet is overlaid on the shape and the dots within the shape are counted. The estimate of area is the number of dots counted multiplied by the area of a single grid square. In some variations, dots that land on or near the boundary of the shape are counted as half of a unit. The dots may also be grouped into larger square groups by lines drawn onto the transparency, allowing groups that are entirely within the shape to be added to the count rather than requiring their dots to be counted one by one. The estimation of area by means of a dot grid has also been called the dot grid method or (particularly when the alignment of the grid with the shape is random) systematic sampling. Perhaps because of its simplicity, it has been repeatedly reinvented.

Application

forestry Forestry is the science and craft of creating, managing, planting, using, conserving and repairing forests, woodlands, and associated resources for human and environmental benefits. Forestry is practiced in plantations and natural stands. ...

cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...

, and

geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...

, the dot planimeter has been applied to maps to estimate the area of parcels of land. In

botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who specialises in this field. The term "botany" comes from the Ancient Greek w ...

and

horticulture Horticulture is the branch of agriculture that deals with the art, science, technology, and business of plant cultivation. It includes the cultivation of fruits, vegetables, nuts, seeds, herbs, sprouts, mushrooms, algae, flowers, seaweeds and no ...

, it has been applied directly to sampled leaves to estimate the average leaf area. In medicine, it has been applied to Lashley diagrams as an estimate of the size of brain lesions. In mineralogy, a similar technique of counting dots in a grid is applied to cross-sections of rock samples for a different purpose, estimating the relative proportions of different constituent minerals.

Theory

Greater accuracy can be achieved by using a dot planimeter with a finer grid of dots. Alternatively, repeatedly placing a dot planimeter with different irrational offsets from its previous placement, and averaging the resulting measurements, can lead to a set of sampled measurements whose average tends towards the true area of the measured shape. The method using a finer grid tends to have better

statistical efficiency In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, needs fewer input data or observations than a less efficient one to achie ...

than repeated measurement with random placements. According to

Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1 ...

, published by

Georg Alexander Pick Georg Alexander Pick (10 August 1859 – 26 July 1942) was an Austrian Jewish mathematician who was murdered during The Holocaust. He was born in Vienna to Josefa Schleisinger and Adolf Josef Pick and died at Concentration camp Theresienstadt, Ther ...

in 1899, the version of the dot planimeter with boundary dots counting as 1/2 (and with an added correction term of −1) gives exact results for

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

s that have the dots as their vertices. According to Blichfeldt's theorem, published by Hans Frederick Blichfeldt in 1914, it is always possible to shift a dot planimeter relative to a given shape without rotating it so that the number of dots within the shape is at least equal to its area. The

Gauss circle problem In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is ...

concerns the error that would be obtained by using a dot planimeter to estimate the area of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

. As its name suggests, it was studied in the early 19th century by

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

. The maximum error is known to be bounded by a fractional power of the radius of the circle, with exponent between 1/2 and 131/208.

Related devices

The dot planimeter differs from other types of

planimeter A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape. Construction There are several kinds of planimeters, but all operate in a similar way. The precise way in whic ...

, which measure the area of a shape by passing a device around its boundary. The Steinhaus longimeter is a similar transparency-based device for estimating the length of curves by counting crossings.

References

{{reflist, refs= {{citation , last = Abell , first = C. A. , journal = Journal of Forestry , pages = 344–345 , title = A method of estimating area in irregularly shaped and broken figures , url = https://cstaecker.fairfield.edu/~cstaecker/files/machines/filer.php?name=dotplanpaperabell.pdf , volume = 37 , year = 1939 {{citation , last = Bellhouse , first = D. R. , doi = 10.2307/2530419 , issue = 2 , journal = Biometrics , jstor = 2530419 , mr = 673040 , pages = 303–312 , title = Area estimation by point-counting techniques , volume = 37 , year = 1981 {{citation , last = Blichfeldt , first = H. F. , author-link = Hans Frederick Blichfeldt , doi = 10.1090/S0002-9947-1914-1500976-6 , doi-access = free , jstor = 1988585 , issue = 3 , journal =

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...

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CiteBank:47270
/ref> {{citation , last = Steinhaus , first = Hugo , author-link = Hugo Steinhaus , issue = 1–2 , journal = Przegląd Matematyczno-Fizyczny , language = pl , pages = 24–29 , title = O mierzeniu pól płaskich , url = https://cstaecker.fairfield.edu/~cstaecker/files/machines/filer.php?name=dotplanpapersteinhaus.pdf , volume = 2 , year = 1924 {{citation , last = Wells , first = David , title = The Penguin Dictionary of Curious and Interesting Geometry , publisher = Penguin Books , year = 1991 , contribution = Pick's theorem , pages = 183–184 {{citation , last = Wood , first = Walter F. , date = January 1954 , doi = 10.1111/j.0033-0124.1954.61_12.x , issue = 1 , journal = The Professional Geographer , pages = 12–14 , title = The dot planimeter, a new way to measure map area , volume = 6

External links

Chris Staecker, Fairfield University Area Dimensional instruments Lattice points Mathematical tools Measuring instruments